nunez otros embodied cognition

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RAFAEL E. NEZ, LAURIE D. EDWARDS and JOO FILIPE MATOS

EMBODIED COGNITION AS GROUNDING FOR SITUATEDNESSAND CONTEXT IN MATHEMATICS EDUCATION

ABSTRACT. In this paper we analyze, from the perspective of Embodied Cognition,why learning and cognition are situated and context-dependent. We argue that the natureof situated learning and cognition cannot be fully understood by focusing only on so-cial, cultural and contextual factors. These factors are themselves further situated andmade comprehensible by the shared biology and fundamental bodily experiences of humanbeings. Thus cognition itself is embodied, and the bodily-grounded nature of cognitionprovides a foundation for social situatedness, entails a reconceptualization of cognition

and mathematics itself, and has important consequences for mathematics education. Afterframing some theoretical notions of embodied cognition in the perspective of moderncognitive science, we analyze a case study continuity of functions. We use conceptualmetaphor theory to show how embodied cognition, while providing grounding for situ-atedness, also gives fruitful results in analyzing the cognitive difficulties underlying theunderstanding of continuity.

1. INTRODUCTION

An important goal of mathematics education is to understand the think-ing involved in doing and learning mathematics. In recent years, it hasbecome widely accepted that the learning and practice of mathematicsare not purely intellectual activities, isolated from social, cultural, andcontextual factors (Lave, 1988; Collins et al., 1989; Cobb, 1994; Confrey,1995). Instead, it has been acknowledged that learning and teaching takeplace, and have always taken place, within embedding social contexts thatdo not just influence, but essentially determine the kinds of knowledgeand practices that are constructed (Lave and Wenger, 1991; Rogoff, 1990;Walkerdine, 1982). Perspectives that focus on the social and contextualnature of knowledge, and that make the embedding situation prominent inthe analysis of cognition, have been labeled as situated. Research andtheoretical frameworks based on a situated approach to cognition insist

that linguistic, social, and interactional factors be included in any accountof subject matter learning, including the learning of mathematics. Thehallmark of this approach is that it considers processes of interaction asbasic and explains individual cognitions and other behaviors in terms of

Educational Studies in Mathematics 39: 4565, 1999. 1999 Kluwer Academic Publishers. Printed in the Netherlands.


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46 RAFAEL E. NUEZ ET AL.

their contributions to interactive systems (Greeno, 1997, p. 15). Lave andWenger make this claim explicit when they state that there is no activitythat is not situated, and when they note the perspectives emphasis on

comprehensive understanding involving the whole person rather than re-ceiving a body of factual knowledge about the world; on activity in andwith the world; and on the view that agent, activity, and the world mutuallyconstitute each other (Lave and Wenger, 1991, p. 33).

These approaches have yielded many important results, and have helpedto move the analysis of learning beyond a narrow focus on individual andinternal cognitive processes. Yet we would argue that the nature of situ-ated learning and cognition cannot be fully understood by attending onlyto contextual or social factors considered as inter-individual processes.Thinking and learning are also situated within biological and experientialcontexts, contexts which have shaped, in a non-arbitrary way, our char-acteristic ways of making sense of the world. These characteristic ways

of understanding, talking about, and acting in the world are shared byhumans by virtue of being interacting members of the same species, co-existing within a given physical medium. The overall aim of this paperis to present elements of a theory which focuses on how human cog-nition is bodily-grounded, that is, embodied within a shared biologicaland physical context, and to examine the ways in which this embodi-ment helps to determine the nature of mathematical understanding andthinking. In particular, we intend to investigate foundational aspects ofsituatedness by bringing in alternative approaches to orthodox cognitivescience, approaches which focus on embodiment. One of our claims isthat the situated cognition perspective, as valuable as it, leaves open im-

portant questions, such as: What is the grounding for situated knowingand learning? What is the basis of social situatedness? We share with thesituated learning approach the belief that knowledge and cognition existand arise within specific social settings, but we go on to ask what it isthat makes possible the mutual intelligibility underlying shared social un-derstandings. Our claim is that the grounding for situatedness comes fromthe nature of shared human bodily experience and action, realized throughbasic embodied cognitive processes and conceptual systems.

When taken seriously, genuine embodiment entails a reconceptualiz-ation of the nature of cognition and of mathematics itself, with corres-ponding implications for teaching (Lakoff and Nez, 1997; forthcoming).A first implication is that we must leave behind the myth of mind-free

mathematics as being about eternal, timeless truths, a legacy of Plato andDescartes. From an embodied perspective, the notion of an objective math-ematics, independent of human understanding, no longer makes sense.


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EMBODIED COGNITION AS GROUNDING FOR SITUATEDNESS 47

Another implication is that we are required to give an account of a mind-based mathematics, including an explanation of its stability and efficacy,in terms of the human bodily-based and situated conceptual systems from

which it arises. Such an account should be useful in understanding prob-lems in the teaching and learning of mathematics, and in designing moreeffective instruction.

In this paper, we will first frame the notion of embodied cognitionwithin an intellectual and theoretical context, and elaborate the relation-ship between situated learning and embodied cognition. Next, we willpresent an example of a mathematical topic continuity that has presen-ted teaching and learning difficulties, as a case study that can be fruitfullyunderstood from an embodied cognition perspective. We close with a dis-cussion of implications and directions for further work in mathematicseducation utilizing the perspective of embodied cognition.

2. FRAMING EMBODIMENT THEORETICALLY AND HISTORICALLY

Early mainstream cognitive science (cognitivism)

The situated learning perspective was welcomed by educational research-ers and theorists as a richer and more appropriate means of addressingcognition than that offered by the formal, cognitivist models of early main-stream cognitive science, which emerged in the 1970s. This latter approach,strongly influenced by the objectivistic tradition of analytical philosophyand by functionalism, focused on learning as a process of individual reas-oning, often explained in computational terms. Researchers in mainstreamcognitive science at that time held that in explaining human cognition, itwas necessary (and sufficient) to posit a level of analysis wholly separatefrom the biological or neurological, on the one hand, and the sociolo-gical or cultural, on the other (Gardner, 1985, p. 6). This separate levelof analysis focused on the individual as a processor of information, andcharacterized reasoning as the manipulation of arbitrary symbols. Underthis view, symbols gain meaning from being associated with an objectivereality, which is modeled in the mind by internal representations corres-ponding, to greater or lesser degrees of accuracy, with that external reality.This view of cognition became (and still is) pervasive in cognitive psy-chology (see for example, Sanford, 1985; Eysenk and Keane, 1992), and

was subsequently adopted by some researchers in mathematics educationseeking a paradigm for understanding mathematical thought.

This school of cognitive science cognitivism maintained the Carte-sian dualism which holds that the mind is an abstract entity, separate from


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and transcending the body. Reasoning (including mathematical thought)is also non-corporeal, timeless, and universal. Concepts, the products ofreasoning, are similarly abstract, and are not limited by physical or bodily

realities. Cognitivism is thus based on objectivism, the doctrine that as-sumes transcendental ontological truths that are independent of human un-derstanding (for details see Nez, 1995). Under this view of knowledge,an objectivist would hold that the Pythagorean theorem, for example, istrue and valid in this and in any universe, irrespective of the existence ofhuman beings. This framing of mind and reasoning is consistent with amodel of cognition as computation, and with a functionalist stance whichholds that is it possible to study the mind purely in terms of the functionsit performs, without seriously considering how the brain and body actuallywork.

The limitations, both theoretical and empirical, of cognitivism have be-come apparent in the 25 years since it became prominent. For example, this

approach has been unable to satisfactorily model or account for everydaycognitive phenomena such as common sense, sense of humor, and naturallanguage understanding. In addition, the information processing modelsthat came out of early mainstream cognitive science bore little resemblanceto the observed processes of real life problem-solving and learning foundeither inside or outside the classroom (Rogoff and Lave, 1984; Lave, 1988;Confrey, 1990; Nunes and Bryant, 1996). Furthermore, the objectivismof mainstream cognitive science was incompatible with the premises ofradical constructivism, which does not assume a pre-determined realitythat is straightforwardly accessed by the observer or learner (Cobb, 1994;Cobb, Yackel and Wood, 1992; Von Glasersfeld, 1990). As a result of these

limitations, many researchers in mathematics education concerned withdevelopmental, social, and cultural factors have rejected cognitive science,assuming that it had little to offer. However, alternative approaches to thescientific study of human mind have emerged within cognitive science it-self, approaches that reject the assumptions of the objectivist, dulist, andfunctionalist school. In these approaches, cognitive processes and con-cepts are not abstract or transcendent, but rather fully embodied, emergentphenomena.

Embodied cognition

In the 1980s, a number of alternative perspectives emerged within cognit-

ive science, originating in different disciplines, but sharing a commitmentto investigating cognition as a physically-embodied phenomenon, realizedvia a process of codetermination between the organism and the mediumin which it exists. Rather than positing a passive observer taking in a


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pre-determined reality, these paradigms hold that reality is constructedby the observer, based on non-arbitrary culturally determined forms ofsense-making which are ultimately grounded in bodily experience.

The term embodiment is used in a number of different ways withincontemporary cognitive science, and these varied uses at times reflect fun-damental theoretical differences. For some, embodiment refers to the phe-nomenological aspects of the human bodily experience (Merleau-Ponty,1945; DiSessa, 1983), and the resulting psychological manifestations(Rosch, 1994). Certain theorists stress the unconscious aspects of bodilyexperience that underlie cognitive activity and linguistic expression (John-son, 1987; Lakoff, 1987). Others focus on the organization of bodily actionunder principles of non-linear dynamics (Thelen and Smith, 1994). Yetothers emphasize the biologico-structural codefinition that exist betweenorganisms and the medium in which they exist, from which cognitionresults as an enactive process (Maturana and Varela, 1987). Along these

lines, some stress the importance of the supra-individual biological pro-cesses that underlie high level cognition (Nez, 1997), and others bring inembodiment as a crucial paradigm in anthropology (Csordas, 1994; Lock,1993). Notions of embodiment are even explicitly used in the design ofresponsive and adaptive non-living systems (Brooks and Stein, 1993) andin structured connectionist computer models of cognitive linguistic activity(Feldman et al., 1996; Regier, 1996).

At a foundational level, our analysis builds on work by Rosch in cognit-ive psychology (Rosch, 1973, 1994; Varela et al., 1991); Edelman (1992)in neuroscience; Maturana and Varela in theoretical biology (Maturana andVarela, 1987) and more explicitly on the work by Lakoff and Johnson in

cognitive linguistics (Lakoff and Johnson, 1980, 1998), and Lakoff andNez in mathematical cognition (Lakoff and Nez, 1997; forthcoming).All these scholars share a focus on the intimate relation between cognition,mind, and living bodily experience in the world, that is, on the ways inwhich complex adaptive behavior emerges from physical experience inbiologically-constrained systems.

Within this paradigm, the knower and the known are codetermined, asare the learner and what is learned. Thus, cognition is about enacting orbringing forth adaptive and effective behavior, not about acquiring inform-ation or representing objects in an external world. The potential of thisperspective for building a more satisfactory account of human thinking isexpressed by Varela, Thompson and Rosch, when they state, If we wish to

recover common sense, then we must invert the representationist attitude[of a pre-given world] by treating context-dependent know-how not asa residual artifact that can be progressively eliminated by the discovery


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of more sophisticated rules, but as, in fact, the very essence of creativecognition (Varela et al., 1991, p. 148).

3. EMBODIMENT AND THE ANALYSIS OF CONCEPTUAL STRUCTURES

Since the concept of embodiment is relatively new within the field of math-ematics education, we would like to clarify our use of the term, and distin-guish it from other notions concerning the role of the physical and concretein mathematics learning. From our perspective, embodiment is not simplyabout an individuals conscious experience of some bodily aspects of beingor acting in the world (e.g., memories of the first time we went skatingor riding in a roller coaster). Embodiment does not necessarily involveconscious awareness of its influence. Nor does embodiment refer to thephysical manipulation of tangible objects (e.g., playing with Cuisinaire

rods or pattern blocks), or to the virtual manipulation of graphical imagesand objects instantiated through technology. Although there is a relation-ship between such experiences and the technical concept of embodiment,an embodied perspective does not constitute a prescription for teaching ina concrete way. Similarly, although embodiment may provide a theoret-ical grounding for understanding the teaching and learning of realistic orcontextualized mathematics, it is not directly concerned with contextu-alization or situatedness in subject matter teaching. Rather, embodimentprovides a deep understanding of what human ideas are, and how they areorganized in vast (mostly unconscious) conceptual systems grounded inphysical, lived reality.

Johnson (1987) offers a nice example of how basic, universal bodilyexperience serves as the grounding for abstract understandings in his dis-cussion of the experience of balance. The experience of balance is partof our everyday life and makes possible our physical experience of theworld as well as our survival in it. The experience of being physically-balanced is so basic and pervasive that we are rarely aware of it. Balancingis an activity we learn with our bodies from very early ages, simply byacting, existing and developing in the world. It is not learned by acquiringabstract rules or algorithms. Moreover, the sensation of balance is so basicthat all homo sapiens no matter when and where they live on earth haveexperienced it. As such, it is one of a class of deep, unconscious, yet per-vasive bodily-based experiences providing a space of commonalties that

makes up the ground for shared human sense-making. At the same time,the meaning of this experience, its working out in cultural expressions suchas language, art, dance, science, and so forth, is both socially-constructedand situated.


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We build up the meaning of balance through the active ongoing ex-perience of bodily equilibrium and loss of equilibrium. Along with thisprocess, we start making sense of related systemic bodily experiences; for

example, the feeling that our fingers are not warm enough, or the mouth istoo dry, and so on. Our understandings of too much, not enough, or outof balance are pre-conceptual, non-formal, and non-propositional. Sense-making is built up in advance of formal or abstract concepts of quantityor balance. The embodied meaning of balance is intimately related toour experience of bodily systemic processes and states of being in theworld, and in particular, to the image-schematic structures that make thoseexperiences coherent and significant for us.

Image-schemata

Image schemata are perceptual-conceptual primitives that allow the organ-

ization of experiences involving spatial relations. Some examples of imageschemata are the containerschema (which underlies concepts like IN andOUT); the source-path-goal schema (TO and FROM); the contactschema;and the verticality schema. Many basic concepts are built on combinationsof these schemata. For instance, the English concept on uses three basicschemata: verticality, contact, and support. Image schemata appear to beuniversal, although in different languages the meanings of the words char-acterizing spatial relations may be composed of different combinations ofthese primitives. For example, not all languages have a single concept likethe English on. In German, the on in on the table is rendered as auf, whilethe on in on the wall, which does not have the support schema, is translatedas an. Thus the two German ons decompose into different combinations ofthe three component image schemata (Lakoff and Nez, forthcoming).

It is important to mention that image-schemata are not static propos-itions that characterize abstract relations between symbols and objectivereality. Rather they are dynamic recurrent patterns which order our actions,perceptions, and conceptions. These patterns emerge as meaningful struc-tures for us mainly through the bodily experience of movement in space,manipulation of objects, and perceptual interactions. As Johnson states,They are a primary means by which we construct or constitute order andnot mere passive receptacles into which experience is poured (Johnson,1987, p. 29).

Abstract concepts such as balancing colors in a picture, balancing

a checking account, or balancing a system of simultaneous equationsare conceptual extensions of the image schemata involved in the bodilyexperience of balance. These extensions occur through concepual map-pings, including the important mechanism known as conceptual metaphor.


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Indeed, it is our thesis that the basis of a great deal of our mathemat-ical knowledge lies in such conceptual mappings. As Lakoff and Nezstate, much of what is abstract in mathematics . . . concerns coordination

of meanings and sense making based on common image-schemata andforms of metaphorical thought. Abstract reasoning and cognition are thusgenuine embodied processes (Lakoff and Nez, 1997, p. 30).

Conceptual metaphor

Conceptual metaphors are mappings that preserve the inferential struc-ture of a source domain as it is projected onto a target domain. Thus thetarget domain is understood, often unconsciously, in terms of the relationsthat hold in the source domain. For instance, within mathematics, Boolean

logic is an extension of the container schema, realized through a concep-tual metaphorical projection of the logic of containers. This metaphor-ical projection preserves the original inferential structure of IN, OUT, andtransitivity, developed originally via physical experiences with actual con-tainers, and later unconsciously mapped to a set of abstract mathematicalconcepts (Lakoff and Nez, forthcoming).

The projections or mappings involved in conceptual metaphors arenot arbitrary, and can be studied empirically and stated precisely. Theyare not arbitrary, because they are motivated by our everyday experience especially bodily experience, which is biologically constrained. Researchin contemporary conceptual metaphor theory has shown that there is an ex-tensive conventional system of conceptual metaphors in every human con-ceptual system. These theoretical claims are based on empirical evidencefrom a variety of sources, including psycholinguistic experiments, gen-eralizations over inference patterns, extensions to novel cases, historicalsemantic change, and the study of spontaneous gestures (Lakoff, 1993).

It has been found that metaphorical mappings are not isolated, but oc-cur in highly-organized systems and combine in complex ways. As withthe rest of our conceptual system, our system of conventional conceptualmetaphors is effortless and lies below the level of conscious awareness(when we consciously produce novel metaphors, we utilize the mechan-isms of our unconscious conventional metaphor system). Unlike traditionalstudies of metaphor, contemporary embodied views dont see conceptual

metaphors as residing in words, but in thought. Metaphorical linguistic ex-pressions thus are only surface manifestations of metaphorical thought (foran extensive discussion of conceptual metaphor theory and mathematics,see Lakoff and Nez, forthcoming).


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Embodiment and situatedness

The fact that embodiment and the mechanisms involved in conceptualmappings specify non-arbitrary links between cognition and experience

helps to answer our question about the grounding for situated knowingand learning. That is, embodiment offers a rationale for the mutual under-standing that exists within social situatedness. As Johnson says,

Meaning is always a matter of human understanding, which constitutes our experience ofa common world that we can make sense of. A theory of meaning is a theory of under-standing. And understanding involves image-schemata and their metaphorical projections. . . These embodied and imaginative structures of meaning have been showed to be shared,public, and objective, in an appropriate sense of objectivity (Johnson, 1987, p. 174).

From this point of view, cognition is neither subjective and isolated unique to an individual nor completely determined by external influ-ences. Conventionalized meaning, although it is never context-free, de-

pends to a great extent on shared image-schemata and conceptual pro-jections, practices, capacities, and knowledge. Meaning is in many wayssocially constructed, but, it is not arbitrary. It is subject to constraintswhich arise from biological embodied processes that take place in theongoing interaction between mutually constituted sense-makers and themedium in which they exist (Nez, 1997). Therefore, it is not surprisingthat cognition and learning are situated. Cognition is embodied; it is bio-logically grounded in individuals who interact with each other; hence it isalso social and cultural.

4. CASE STUDY: CONTINUITY OF FUNCTIONS

It is widely accepted that teaching and learning the concept of continuityof a function, so important for calculus, is a difficult task (Tall and Vin-ner, 1981; Robert, 1982; Nez, 1993; Kitcher, 1997). The question thenis, why is this the case? Is continuity per se a difficult concept? In thissection we would like to illustrate how embodied cognition offers fruitfulanswers to these questions, by utilizing the tools of cognitive linguistics toanalyze the concept (for details see Lakoff and Nez, 1997, and Nezand Lakoff, 1998).

Let us start taking a look at what textbooks say about continuity of afunction. Here is a citation from a typical textbook introducing the concept:

In everyday speech, a continuous process is one that proceeds without gaps or inter-ruptions or sudden changes. Roughly speaking, a function y = f(x) is continuous if itdisplays similar behavior, that is, if a small change in x produces a small change in thecorresponding value f(x) ...Up to this stage, our remarks about continuity have been


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rather loose and intuitive, and intended more to explain than to define (Simmons, 1985,p. 58).

Later in the same text, one finds what is called the rigorous, formal and

definitive definition of continuity of a function, namely:A function f is continuous at a number a if the following three conditions are satisfied:1. fis defined on an open interval containing a,2. limxa f(x) exists, and3. limxa f(x) = f(a).Where limxa f(x) (the limit of the function at a) is defined as:Let a function f be defined on an open interval containing a, except possibly at a itself,and let L be a real number. The statement

limxa f(x) = L

means that for every > 0, there exists a > 0, such that

if 0 < |x a| < , then |f(x) L| < .

This definition of continuity of a function also called the Cauchy-Weier-strass definition is said to be, and taught as, the definition that capturesthe essence of what continuity is. It is considered, and taught as, super-ior and more precise than the so-called intuitive and informal one.Moreover, as is evident in the text cited above, this definition intends moreto define than to explain.

So far, this is the standard (and from our point of view, misleading,non-situated, disembodied) story. Let us step back, and carefully analyzewhat is going on, cognitively, when considering the statements and ideasinvolved in the two definitions.

The two definitions of continuity

The informal/intuitive definition that characterizes a continuous processas one that proceeds without gaps or interruptions or sudden changeswas used by such eminent mathematicians as Newton and Leibniz in the17th century. Euler characterized a continuous function as a curve de-scribed by freely leading the hand. This definition involves cognitive con-tents such as motion, flows, processes, change in time, and wholeness.These cognitive contents are the result of conceptual extensions from bod-ily grounded image-schemata and conceptual mappings that are naturalto the human conceptual system. They are built on, among others, source-path-goal schemata, fictive motion metaphors, and basic conceptual blends(for details see Lakoff and Nez, 1997). For these reasons, the textbook

previously mentioned is correct in referring to this idea as occurring ineveryday speech. What Newton, Leibniz, and Euler did was simply (and,probably, unconsciously) apply the inferential structure of the everydayunderstanding of motion, flow, and wholeness, to a specific domain of


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human understanding: functions and variations. For the purposes of thiscase study, we will call this concept natural continuity.

The Cauchy-Weierstrass definition on the other hand, involves radically

different cognitive content. It implicitly denies motion, flow and whole-ness, dealing exclusively with static, discrete, and atomistic elements, whichare conceptual extensions of rather different cognitive primitives, such aspart-whole schemata and container schemata. The point is that, cognit-ively speaking, these two definitions are radically different; yet, per se,neither is superior to the other. Although it is true that the so-called rigor-ous definition deals better with complex and pathological cases (such asf(x) = x sin1/x) for certain purposes, it is not because it captures betterthe essence of continuity. Within an embodied, non-objectivist cognitivescience, there is no transcendental essence of a concept, even in mathem-atics (Edwards and Nez, 1995). It does so simply because it is built on adifferent collection of bodily grounded conceptual mappings that happen

to deal well with both the prototypical cases of functions encountered priorto the 19th century (e.g., f(x) = sin x; or f(x) = 1/x), as well as with theso-called pathological cases. This is the source of its utility and efficacy.

For the purposes of this article, the pedagogical problem can be sum-marized as follows: students are introduced to natural continuity usingconcepts, ideas, and examples which draw on inferential patterns sustainedby the natural human conceptual system. Then, they are introduced toanother concept Cauchy-Weierstrass continuity that rests upon radic-ally different cognitive contents (although not necessarily more complex).These contents draw on different inferential structures and different entail-ments that conflict with those from the previous idea. The problem is that

students are never told that the new definition is actually a completely dif-ferent human-embodied idea. Worse, they are told that the new definitioncaptures the essence of the old idea, which, by virtue of being intuit-ive and vague, is to be avoided. This essence is usually understood assituation-free, that is, independent of human understanding, social activity,and philosophical enterprises.

Embodied cognition analysis of the two concepts

Let us analyze, from the perspective of embodied cognition, why these twoconcepts of continuity natural and Cauchy-Weierstrass are cognitivelyso different. Although it is not within the scope of this article to provide

a full cognitive analysis of these two ideas (for a complete analysis, seeLakoff and Nez, forthcoming), we will present several relevant aspectsfor the purpose of a deeper comparison. In particular we will focus on thefact that the term continuity, as used in mathematics, can refer to three


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distinct ideas. One is natural continuity (as in the informal, intuitivedefinition) and the other two (implicit in the rigorous Cauchy-Weierstrassdefinition) are Gaplessness (for lines as sets of points) and Preservation of

Closeness (for functions).

Natural continuity:

The following are some essential features of a continuous function accord-ing to natural continuity:

a) the continuous function is formed by motion, which takes place overtime.

b) there is a directionality in the function.c) the continuity arises from the motion.d) since there is motion, there is some entity moving (in Eulers version,

the hand).e) the motion results in a static line with no jumps.f) the static line that results has no directionality.

What is the source of these notions of motion, staticness, and direction-ality? From the perspective of embodied cognition, we conceive of themobile and static aspects of a continuous curve via the activation of aneveryday human conceptual process: the fictive motion metaphor (Talmy,1988). This metaphor can be summarized as follows:

A Line IS The Motion of a Traveler tracing that line.

Examples of this mapping are abundant in everyday language:

Highway 80 goes to Sacramento. Just before Highway 24 reaches Walnut Creek, it goes through the

Caldecott Tunnel. After crossing the bay, Highway 80 reaches San Francisco.

In these cases a highway, which is a static linear object, is conceptualizedin terms of a traveler moving along the route of the highway. Using thesame cognitive mechanism, we can speak, in mathematics, of a functionas moving, growing, oscillating, approaching values, and reaching lim-its. It is worth noting that this way of speaking is not limited to studentsbut includes professional mathematicians as well. Formally speaking, thefunction does not move, but cognitively speaking, under this metaphor, it

does and that is what matters in terms of understanding.These embodied natural and everyday human cognitive mechanisms are

the ones that make possible the intuitive dynamic and static conceptualiz-ation of a continuous function. As in Eulers characterization, continuity is


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characterized by motion in the Fictive Motion metaphor. Using this meth-odology we can give a precise cognitive account of Eulers intuitive notionof continuity for a function in terms of elements of ordinary embodied

human cognition, showing how mathematical ideas are constituted out ofordinary bodily grounded ideas.

Cauchy-Weierstrass continuity:

The Cauchy-Weierstrass definition was motivated by complex mathem-atical objects that mathematicians first encountered in the 19th century,and emerged from three important intellectual movements of that time:the arithmetization of analysis; the set-theoretical foundations movement;and the philosophy of formalism. These movements were separate in theirgoals, but complementary in their effects on the development of mathemat-ics. All of them required conceptualizing lines, planes, and n-dimensional

spaces as sets of points.The Cauchy-Weierstrass definition requires a series of cognitive prim-itives, also embodied in nature, but different from the ones used to con-ceptualize natural continuity. There are at least three relevant conceptualmetaphors that combine their inferential structures in a systemic way togive an extremely powerful mathematical tool. These metaphors are:

A Line IS a Set of Points Natural Continuity IS Gaplessness Approaching a Limit IS Preservation of Closeness Near a Point

A Line IS a Set Of Points

In general terms, there are two importantly different ways of conceptual-izing a line:1) A holistic one, not made up of discrete elements, where a line is abso-lutely continuous and points are locations on a line. In this sense, a lineis an entity distinct from the points, that is, locations on that line, just as ahighway is a distinct entity from the locations on that highway. Lines, fromthe perspective of our everyday geometric intuition, are natural continua inthis sense.2) A Line Is A Set Of Points. According to this metaphor, the points arenot locations on the line, but rather they are entities constituting the line.

The first characterization is congruent with natural continuity and the

second with Cauchy-Weierstrass definition. The distinction between thesetwo ways of conceptualizing lines (and hence planes and n-dimensionalspaces) has been crucial throughout the history of mathematics, and thefailure to distinguish between them has led to considerable confusion. Both


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conceptions are natural, in that both arise from our everyday conceptualsystem. Neither is right or wrong per se; however, they have very dif-ferent cognitive properties, and provide different inferential structure. It

is this fact that should be taken into account in the teaching and learningprocess.

Natural Continuity IS Gaplessness

According to our everyday intuition, a line constitutes a natural continuum.As we move along a line, we go through point-locations. When we movecontinuously along a line from a location A to a location B, we go throughall point-locations on the line between A and B, without skipping overany, that is, without leaving any gaps between the point-locations. In thiscase we will say that the collection of point-locations between A and B isgapless when the line segment AB is naturally continuous.

The metaphor underlying the Cauchy-Weierstrass definition identifiesthe point-locations on a line as constituting the line itself. Such a meta-phorical line is not a natural continuum, but only a set of points. Whena naturally continuous line segment is conceptualized as a set of points,that set of points will be gapless. Thus, in this specific situated conceptualcontext, the metaphor A Line Is A Set Of Points entails the metaphor:

Natural Continuity IS Gaplessness.

Therefore, a line conceptualized as a set of points cannot be cognitively naturally continuous but only gapless. This terminology thus distinguishestwo distinct ideas, based on different cognitive mechanisms, that have

previously both been called continuity.

Approaching a Limit IS Preservation of Closeness Near a Point

In Cauchy-Weierstrass definition of limit there is no motion, no time, andno approach. Instead, there are static elements. The definition calls for agapless open interval of real numbers; there are no lines and no pointsand no surfaces in that metaphorical ontology for the Cartesian plane. Theplane itself is a made up of a set of pairs of real numbers. The gapless-ness of the set of real numbers in the open interval is Cauchy-Weierstrassmetaphorical version replacing the natural continuity of the intuitive linein Newtons geometric idea of a limit.

The idea of the function f approaching a limit L as x approaches a isreplaced by a different idea (in order to arithmetize approaching avoidingmotion), that is, preservation of closeness near a real number: f(x) is arbit-rarily close to L when x is sufficiently close to a. The epsilon-delta condi-


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tion expresses this precisely in formal logic. What the Cauchy-Weierstrassapproach does is to provide a new metaphor:

Approaching A Limit IS Preservation Of Closeness Near A Point.

When Cauchy-Weierstrass define continuity for a function, they cannotmean cognitively the natural continuity assumed by Newton for or-dinary lines, that is, natural continua. Again, they must use conceptualmappings (metaphors) that allow them to reconceptualize geometry (hol-istic lines) using arithmetic (discrete numbers). Just as they needed a newmetaphor for approaching a limit, they needed a new metaphor for con-tinuity of a function. They characterize this new metaphor in two steps:first at a single arbitrary real number, and then throughout a (gapless)interval. Their new metaphor for continuity uses the same basic idea astheir metaphor for a limit: preservation of closeness. Continuity at a real

number is conceptualized as preservation of closeness, not just near a realnumber but also atit. Continuity of a function throughout an interval is thuspreservation of closeness near and at every real number in the interval.

What is precise in the Cauchy-Weierstrass definition?

Textbooks and curricula lead students to believe that it is the epsilon-deltaportion of these definitions that constitutes the rigor of this arithmetizationof analysis. Moreover, they are led to believe that it is this aspect thatcaptures the essence of what continuity is. As we see, not only this isnot true, but the epsilon-delta aspect of the definition actually plays a farmore limited role. The epsilon-delta aspect accomplishes only a precisecharacterization of the notion correspondingly. This notion occurs in thedynamic definition of a limit, where the values off(x) get correspondinglycloser to L as x gets closer to a. This is the only vagueness that is madeprecise by the epsilon-delta definition.

Another interesting element in the Cauchy-Weierstrass definition is therole played by the idea of gaplessness. The Cauchy-Weierstrass formu-lates the definition of continuity with the explicit condition that the func-tion is defined over an open interval. It assumes this open interval to begapless. Since gaplessness was the way to metaphorically conceptualizecontinuity on the real line, it assumes a gapless (i.e., continuous) inputto the function. What this definition really shows is that (1) when these

metaphors hold, especially when lines are metaphorically conceptualizedas sets of real numbers, and (2) when the input of the function is gapless,and (3) when the function preserves closeness, then (4) the output is alsogapless.


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Why has it been widely accepted that Cauchy-Weierstrass definitionof preservation of closeness was instead a definition of continuity? Theanswer is that it has been assumed, falsely, that Cauchy-Weierstrass meta-

phors capture the essence of continuity because they deal effectively, forthe purposes of the arithmetization program, with prototypical and patho-logical cases. Given the metaphor that a line is a set of real numbers, thennatural continuity can only be conceptualized metaphorically as gapless-ness. Since Cauchy-Weierstrass open interval condition guaranteed thatthe inputs to the function are always gapless, it is no surprise that preser-vation of closeness for a function with a gapless input guarantees a gaplessoutput. If the input is metaphorically continuous (that is, gapless), thenthe output is going to be metaphorically continuous (gapless). Since themetaphors are mostly realized through unconscious processes, and theyfit the prototypical cases, they are not noticed as being metaphorical orcontroversial in any way. Furthermore, since the open interval condition

hid the continuity (gaplessness) required in the input, Cauchy-Weierstrassdefinition appeared even to the originators to be a definition of continuity,when in fact, all it did was guarantee that a gapless input for a functiongives a gapless output.

5. DISCUSSION

As a discipline, mathematics education is concerned not only with creatingeffective means and methods of instruction, but with understanding whycertain methods are effective and others are not, and with larger ques-tions about the nature and development of mathematical knowledge. Ouranswers to these questions, and even the ways we choose to investigatethem, are strongly influenced by our implicit or explicit conceptualizationabout the nature of human thought, and about mathematics itself. Whenmathematics is conceived of as an external realm of objective truths, to bediscovered through the application of rational thinking, then the invest-igation of mathematics learning focuses on accurate mappings, models,and internal representations of mathematical entities and relationships. If,on the other hand, mathematics is conceived as a product of adaptive hu-man activity in the world, shared and made meaningful through language,but based ultimately on biological and bodily experiences unique to ourspecies, then mathematics education must take a different approach. New

practices in mathematics education, from classroom teaching to scientificresearch and curriculum design, should emerge that present mathematicsas a genuine mind-based activity with all its embodied peculiarities andbeauty.


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Through the analysis of the idea of continuity, we have seen that cer-tain ways of talking and thinking about mathematics can be misleading,with unfortunate pedagogical consequences. These consequences can arise

when we ignore how our conceptual system works, implicitly assuming theexistence of a mind-free mathematics. We propose that one importantsource of pedagogical problems in mathematics education are the philo-sophical foundations that have dominated our view of mathematics (ob-jectivism, platonism, formalism). These philosophical commitments arenecessarily (if unintentionally) transmitted in the teaching process, whichcan lead to the teaching of definitions and supposed eternal truths thatcapture mathematical essences, rather than mind-based, embodied, humanforms of sense-making. The fundamental conceptual error underlying thiskind of teaching is the idea that intuition can be replaced by rigor in orderto eliminate vagueness. Not only is this not possible, but it is not necessaryfor effective learning (c.f., Smith et al., 1993/94). If one studies natural,

situated, spontaneous, everyday thoughts and intuitions from an embod-ied cognitive perspective, one finds that they are not at all vague. Thetools provided by the embodied cognition approach allow one to character-ize precisely how the inferential structure of everyday bodily experience,which underlies intuition, is mapped onto more abstract domains.

Basic mathematical ideas show an impressive stability over hundreds,sometimes thousands of years. For this to happen requires, on the onehand, a common set of neural and bodily structures with which to contructmathematical concepts. On the other hand, it requires that this conceptualconstruction make use of the most commonplace of everyday experiences,such as motion, spatial relations, object manipulation, space, and time.

The study of the conceptual structure of mathematics from an embodiedpoint of view shows how mathematics is built up out of such informal,everyday experiences and ideas. For this reason, mathematics cannot beconceived as a pure and abstract discipline. Our mathematical concep-tual system, like the rest of our conceptual system, is grounded in ourbodily functioning and experiences. Seen from this perspective, situatedcognition is not about situating mind-free truths in meaningful contexts,but rather about examining how the human creation of mathematics arisesfrom sense-making which is not arbitrary precisely because it is bodilygrounded.

This view has important entailments for mathematics education. Ratherthan looking for better ways to help students learn rigorous definitions

of pre-given mathematical ideas, we need to examine the kinds of un-derstanding and sense-making we want students to develop. We shouldlook at the everyday experiences that provide the initial grounding for


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the abstractions that constitute mathematics. This is not necessarily aneasy undertaking, since the grounding structures are often unconsciousand taken-for-granted. At times, this grounding can be found in imme-

diate physical experience, as in the case of work with early arithmetic,space, size, and motion. At other times, the grounding for a mathemat-ical idea takes place indirectly, through a chain of conceptual mappingswhose nature may be obscured by conventional language, but which canbe revealed by utilizing the analytic tools of contemporary embodied cog-nitive science. In either case, what is important is to re-examine mathem-atical ideas in order to create instruction that complements the ways ourconceptual systems naturally work.

In addition, we should provide a learning environment in which math-ematical ideas are taught and discussed with all their human embodiedand social features. Students (and teachers) should know that mathematicaltheorems, proofs, and objects are about ideas, and that these ideas are situ-

ated and meaningful because they are grounded in our bodily experienceas social animals. Providing an understanding of the historical processesthrough which embodied ideas have emerged can support this aim. Thisdoes not mean simply presenting a few names and dates as a preludeto teaching the real mathematics. It means talking about the motiva-tions, zeitgeist, controversies, difficulties, and disputes that motivated andmade possible particular developments in mathematics. Pierpont (1899),for example, provides excellent material for appreciating the controversiessurrounding intuition, the concept of continuity, and the arithmetizationprogram at the turn of the century.

In this paper, we have attempted to provide an overview of the essential

elements of a theory of embodied cognition, and to apply this relativelynew framework to the analysis of mathematical thought and learning. Thisframework is extremely rich, and systematic work on the analysis of math-ematical thought is only beginning to take place. We presented a briefaccount of one such analysis, in order to illustrate the potential of thisframework. We also addressed the relationship between theories whichemphasize the socially-situated nature of cognition and the embodied cog-nition approaches. From our perspective, there is no contradiction betweenthese approaches; rather, an understanding of the fundamental embodimentof cognition helps us to see how human beings are able to construct mutualunderstandings through social interaction. Since we, human beings, are allliving physical creatures, acting within the same medium and sharing a

basic biological heritage, we naturally experience the world in fundament-ally similar ways. The conceptual structures which emerge in the humanmind to make sense of our bodily experiences provide the raw material


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for the construction of shared communication through language, and, sub-sequently, the shared construction of meanings. Thus, our understandingsof the world, and of mathematics, may be socially and culturally situated,

but is the commonalties in our physical embodiment and experience thatprovide the bedrock for this situatedness.

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RAFAEL E. NEZ

Institute of Cognitive Studies,

Berkeley, CA 94720,

U.S.A.

E-mail: [email protected]

LAURIE D. EDWARDS

School of Education,St. Marys College of California,

Moraga, CA 94575,

U.S.A.

JOO FILIPE MATOS

Department of Education,

University of Lisbon,

Portugal


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